1

Chapter 2 :

Fundamental parameters of antennas

• From Radiation Pattern

– Radiation intensity

– Beamwidth

– Directivity

– Antenna efficiency

– Gain

– Polarization

• From Circuit viewpoint

– Input Impedance

2

Chapter 2 : Topics (2)

• Antenna effective length and effective area

• Friis transmission equation

• Radar range equation

• Antenna temperature

3

Definition of Radiation Pattern

• Once the electromagnetic (EM) energy

leaves the antenna, the radiation pattern

tells us how the energy propagates away

from the antenna.

• Definition :

Mathematical function or a graphical

representation of the radiation properties

of an antenna as a function of space coordinates

4

Radiation Pattern Example

5

Radiation Pattern (1)

• Can be classified as:

– Isotropic, directional and omnidirectional

• Isotropic: Hypothetical antenna having equal

radiation in all directions

• Directional: having the property of

transmitting or receiving EM energy more

effectively in some directions than others

• Omnidirectional: having an essentially

nondirectional pattern in a given plane and a

directional pattern in any orthogonal plane

6

Radiation Pattern (2)

• Principal patterns (or planes):

– E-plane : the plane containing the electric field

vector and the direction of maximum radiation

– H-plane : the plane containing the magnetic

field vector and the direction of maximum

radiation

7

Radiation Pattern (3)

Omnidirectional

Eθ

Hφ

8

Field Regions

DR1

R2

Reactive near

field region

Radiating near field

region (Fresnel zone)

Far field region

(Fraunhofer zone)

D=largest antenna

dimension

9

Reactive Near Field Region

• Region surrounding the antenna, wherein the

reactive field predominates

]62.0,2

max[

2 :antenna) (smallFor

62.0:For

3

3

DR

RD

DRD

Angular field distribution depends on distance from

antenna

10

Radiating Near Field Region

• Region between reactive near-field and far-

field regions (Fresnel zone)

]62.0,2

max[]2

,3max[

23 :antenna) (smallFor

62.02

:For

32

32

DR

D

RD

DR

DD

Radiation fields predominate but angular field

distribution still depends on distance from antenna

11

Far Field Region

• Region where angular field distribution is

essentially independent of the distance from

antenna (Fraunhofer zone)

]2

,3max[

3 :antenna) (smallFor

2:For

2

2

DR

RD

DRD

12

Change of antenna amplitude

pattern shape

13

Spherical coordinate and

Solid Angle : Steradian• Measure of solid

angle: 1 steradian =

solid angle with its

vertex at the center

of a sphere of radius

r that is subtended

by a surface of area

r2

Quiz: What’s the solid angle subtended by a sphere?

anglesolidsin

;sin 22

ddd

drddrdA

14

Radiation Power Density

• Poynting vector = Power density

• Total power:

HHHHEEEEWWWWrrr

[A/m]Intensity field magnetic ousinstantane :

[V/m]Intensity field electric ousinstantane :

][W/m vector Poynting ousinstantane : 2

HHHH

EEEE

WWWW

r

r

r

SS

dand ˆWWWWWWWWrr

sP

][m surface closed theof area malinfinitesi :

surface the tonormalr unit vecto :ˆ

[W]power totalousinstantane :

2da

n

P

15

Radiation Power Density (2)• For time-harmonic EM fields

• Poynting vector

• Time average Poynting vector (average power

density or radiation density)

]),,(Re[);,,( tjezyxtzyx

Err

EEEE

]),,(Re[);,,( tjezyxtzyx

Hrr

HHHH

]Re[2

1]Re[

2

1 2* tje

HEHErrrrrrr

HHHHEEEEWWWW

speak valuerepresent fields, because appears2

1

]Re[2

1)];,,([),,( *

HE

HEW

rr

rrrr avav tzyxzyx WWWW

16

Radiation Power Density (3)

• Average power radiated power

]W[sinˆ)sin

ˆ(

ˆ

0

22

0 0

2

20 Addrrr

Ar

danPS

avrad

Wr

SS

av

S

avavrad ddandPP sHEWW ]Re[2

1ˆ *

rrrrs

Example 2.1: The average power density is given by

The total radiated power becomes

]m/W[sin

ˆˆ 2

20r

ArWr rav

W

r

17

Radiation Power Density (4)

• For an isotropic antenna

The power density is then given by

]m/W[4

ˆˆ 2

200r

PrWr rad

W

r

]W[4sinˆ)](ˆ[ 0

22

0 0

2

0

0

WrddrrrWr

dPS

rad

sWrr

18

Radiation Intensity

• Definition : The power radiated from an

antenna per unit solid angle

2

0 0sin ddUUdPrad

Total power can be given by

radWrU2

][W/mdensity radiation :

angle] solid[W/unit intensity radiation :

2

radW

U

19

Radiation Intensity (2)

• Radiation intensity is related to the far-zone

electric field of antenna

]|),(||),([|2

1

]|),,(||),,([|2

|),,(|2

),(

22

222

22

ooEE

rErEr

rr

U

Er

space) freein 377 ( medium theof impedance intrinsic :

antenna theof components field-electric zone-far :,

),( antenna theofintensity field-electric zone-far :),,(

η

EE

r

er

jkro

EErr

20

Radiation Intensity (3)

000 4 UdUdUPrad

]W[sin

sin

0

22

0 0

2

0

2

0 0

AddA

ddUPrad

sin0

2AWrU rad

Example 2.2: The radiation intensity is given by

The total radiated power becomes

For an isotropic antenna

40

radPU

21

Beamwidth

• Beamwidth is the angular separation between two identical points on opposite site of the pattern maximum

• Half-power beamwidth (HPBW): in a plane containing the direction of the maximum of a beam, the angle between the two directions in which the radiation intensity is one-half value of the beam

• First-Null beamwidth (FNBW): angular separation between the first nulls of the pattern

22

Beamwidth (2)

23

Beamwidth (3)

)20,0(cos)( 2 U

4)707.0(cos

707.0cos5.0cos|)(

1

2

h

hhU

Example 2.3: The normalized radiation intensity of an

antenna is represented by

The angle θh at which the function equal to half of its maximum can be found by

Since the pattern is symmetric with respect to the

maximum, HPBW = 2 θh = π/2

Likewise, FNBW = 2θn = π since2

)0(cos0|)( 1

nnU

24

Directivity

• Ratio of radiation intensity in a given direction

from the antenna to the average radiation intensity

less)-(dimension4),(

),(0 radP

U

U

UD

Note that the average radiation intensity equals to the

radiation intensity of an isotropic source.

25

Directivity (2)

AdU

UD

dU

UD

4

')','(4

')','(

),(4),(

'

maxmax

'

'

')','( dUPrad

ΩA is called beam solid angle, and is defined as “solid

angle through which all the power of the antenna would

flow if its radiation intensity were constant and equal to

Umax for all angles within ΩA

Since

max

' max

')','(

UPdU

UAradA

Dmax: maximum directivity

26

Directivity (3)

• If the direction is not

specified, it implies the

directivity of maximum

radiation intensity

(maximum directivity)

expressed as

less)-(dimension4 max

0

max0max

radP

U

U

UDD

27

Directivity (4)

2

0

2 sinAWrU r

]W/m[sin

ˆˆ 2

2

2

0r

ArWrW rav

2

0 00

2

03

8sinsin AddAUdPrad

Example 2.4: The radial component of the radiated power density of an infinitesimal linear dipole is given by

where A0 is the peak value of the power density. The radiation intensity is given by

The maximum radiation is directed along θ = π/2 and Umax = A0.

The total radiated power is given by

Thus,

22

0max

0 sin5.1sin and 2

34 DD

P

UD

rad

28

Antenna Efficiency

• The overall antenna efficiency take into the

following losses:

– Reflections because of the mismatch between the

transmission line and the antenna

– Conduction and dielectric losses

)||1( 2

0 cdrcd eeee

inalinput term at thet coefficien reflection voltage:

esefficienci dielectric ,conduction :,

efficiencyradiation antenna :

efficiency (mismatch) reflection :

efficiency total:0

dc

dccd

r

ee

eee

e

e

29

Gain

• It takes into account the efficiency of the antenna as well

as its directional properties. (Directivity only measures

directional properties.)

inP

UG

),(4),(

loss)power dielectric and Ohmic:(

)||1(

antenna; input toPower :

2

loss

lossradoin

in

P

PPPP

P

Gain : ratio of radiation intensity in a given direction to the

average radiation intensity that would be obtained if all the

power input to the antenna were radiated isotropically

30

Gain (2)

• Using ecd, Prad= ecdPin and

),(),(

4),(

DeeP

UG cdcd

rad

Relative gain: ratio of power gain in a given direction to the

power gain of a reference antenna in the same direction. The

power input must be the same for both antennas. If the reference antenna is a lossless isotropic source, then

source) isotropic (lossless

),(4),(

inP

UG

31

Gain (3)

• Absolute gain takes into account impedance mismatch

losses at the input terminals in addition to losses within

antenna

),()||1(),(

4

)||1(),(

4),(

4),(

0

2

2

DeeP

U

P

U

P

UG

cd

rad

ino

abs

32

Polarization• Property of an EM wave describing the time varying

direction and relative magnitude of the electric field.

The figure traced as a function of time by the tip of

the electric field and the sense in which is traced, as

observed along direction of propagation.

0, where)(

,)(Let

yoxo

jkzj

yoy

jkzj

xox

EEeeEzE

eeEzE

y

x

);(ˆ);(ˆ);( tzytzxtz yx EEEEEEEEEEEE r

)cos();(

)cos();(

yyoy

xxox

kztEtz

kztEtz

EEEE

EEEE

x

y

z

dependence timetje

Wave propagating in –z

direction

33

Polarization (2)

0or 0(i) yoxo EE

z

22

yoxo EE

A. Linear Polarization

Example

20,tan

,...2,1,0 where

(ii)

1

xo

yo

xy

E

E

n

n

δ, γ determine polarization state

or

34

Polarization (3)

4/)1(tan (i) 1 oyoxo EEE

B. Circular Polarization

,...2,1,0 where

CCW/LCP;2

12n

CW/RCP;2

12n

(ii)

n

xy

and

Note that the sense of rotation is observed along the direction of propagation.

35

36

Polarization (4)

2/,4/

)sin(

)2/cos();(

)cos();(

xo

xoy

xox

kztE

kztEtz

kztEtz

EEEE

EEEE

Example: RCP

z

37

Polarization (5)

C. Elliptic Polarization

A wave is elliptically polarized if it is not linearly or

circularly polarized.

Linear and circular polarization are special cases of elliptic polarization.

To have elliptic polarization:

1. Field must have two orthogonal linear components.

2. The two components can be of the same or different magnitude.

38

Polarization (6)

yoxo

xy

EEn

n

, FOR,...2,1,0 where

CCW/LEP;0

CW/REP;0

2 (ii)if

C. Elliptic Polarization

yoxo

xy

EEn

AND,...2,1,0 where

CCW/LEP;2

12n

CW/REP;2

12n

(i)if

39

Polarization (7)

AR1;AxisMinor

AxisMajor (AR) Ratio Axial

cos

2tan

2

1

2 22

1

yoxo

yoxo

EE

EE

40

Polarization Loss Factor (PLF)

• Electric field of incoming wave

• Electric field of receiving antennaiww EE

r

less)(dimension

|cos||ˆˆ|PLF 22

paw

aaa EEr

on vectorpolarizati:ˆ

wave theofr unit vecto:ˆ

where

a

w

a

w

41

PLF example

jkzj

oy

jkzj

ox eeEEeeEE yx ;

2

ˆˆˆwhere

2ˆ)ˆˆ()ˆˆ( 2/

yjx

eEeEyjxeeEyEx

w

jkz

ow

jkz

o

jkzj

oo

Er

*ˆˆwa

LCP wave: δ=-π/2,φx=0φy=-π/2

dB 012

ˆˆ

2

ˆˆPLF

2

yjxyjx

If the antenna is also LCP,

If the antenna is RCP,2

ˆˆˆ

yjxa

02

ˆˆ

2

ˆˆPLF

2

yjxyjx

42

Input Impedance

Generator

Antenna

Terminal

a

b

dohmicloss

loss

r

lossrA

RRR

R

R

RRR

)dielectric(ohmic, resistance Loss :

resistanceRadiation :

Transmitting case

Impedance presented by the antenna at its terminal

speak value are ,

)(,

gg

Agggggg

IV

ZZIVjXRZ

][)()()( AAA jXRZ

gAgA XXRR ,Maximum power delivered to the antenna occurs when conjugate

matched:

43

Input Impedance (2)

2

2

2

)(8

||||

2

1

2 lossr

rg

rgrad

A

g

gRR

RVRIP

R

VI

2

2

2

)(8

||||

2

1

lossr

lossg

lossglossRR

RVRIP

lossradg PPP

lossr

g

gggRR

VRIP

1

8

||||

2

12

2

When conjugate matched:

NOTE:

Radiated Power

Power loss to heat

Power loss in Rg

44

Input Impedance (3)

slossradin PPPP2

1

g

g

A

g

A

g

gggsR

V

R

V

R

VVIVP

4

||

4

||

22

1]

2

1Re[

22*

*

ing PP

lossr

r

in

radcd

RR

R

P

Pe

1,,00 If cdinradlossloss ePPPR

Power supplied by generator when conjugate matched:

antenna radiation

efficiency

sg PP2

1

45

Receiving Antenna

speak value are ,

8

||||

2

1

2

22

TT

T

TTTT

T

TT

IV

R

VRIP

R

VI

ATlossrAT XXRRRR ,

condition matched conjugateunder Receiving case

Impedance presented by the antenna at its terminal

TTTAAA jXRZjXRZ ,

2

22

)(8

||||

2

1

lossr

rTrTscatt

RR

RVRIP

Power delivered to

load

Power scattered or re-radiated

46

Receiving Antenna (2)

T

TTTc

R

VIVP

4

||]Re[

2

1 2*

2

22

)(8

||||

2

1

lossr

lossTlossTloss

RR

RVRIP

cT PP2

1

clossscatt PPP2

1

Tlossscattc PPPP

Power supplied by generator when conjugate matched:

captured/collected power

lossscattT PPP Power lost to heat

47

Antenna equivalent Area

• Used to describe the power capturing characteristics of

an antenna when a wave impinges on it

),(direction fromdensity flux power Incident

antenna receiving of lsat termina availablePower

),(direction in (aperture) area effective),(

eA

]m[2

|| 22

i

TT

i

Te

W

RI

W

PA

aveincident w ofdensity power :

load todeliveredpower :

i

TT

W

RP

48

Antenna Equivalent Area (2)

22

2

)()(2

||

TATlossr

T

i

Te

XXRRR

R

W

VA

2

2

)(8

||

lossr

r

i

Ts

RR

R

W

VA

lossri

T

Ti

Tem

RRW

V

RW

VA

1

8

||

8

|| 22

Under conjugate matched condition:

power radiated-reor scattered the toequal is density,

powerincident by multipliedn which whearea, scattering:

load todeliveredpower toequal is density,power

incident by the multipliedn which whearea, effective :e

s

T

A

R

A

Maximum effective area

49

Antenna Equivalent Area (3)

2

22

)(8

||

4

||

lossr

Tlossr

i

T

Ti

Tc

RR

RRR

W

V

RW

VA

2

2

)(8

||

lossr

loss

i

Tloss

RR

R

W

VA

antennaby capturedpower total the toequal is density,

powerincident by multipliedn which whearea, Captured:

load todeliveredpower toequal is density,

powerincident by the multipliedn which whearea, loss :loss

c

loss

A

R

A

Under conjugate matched condition:

lossse AAA

losssec AAAA

50

Antenna Equivalent Area (4)

ri

Tem

RW

VA

8

|| 2

2

2E

Wi

ElVT

Example 2.5: a uniform plane wave is incident upon very

short dipole, whose radiation resistance is Rr=80(πl/λ)2.

Assume that Rloss = 0, the maximum effective area reduces to

Since the dipole is very short, the induced current can be

assumed to be constant and of uniform phase. The induced

voltage is

22

2222

2

119.08

3

)/80)(2/(8

)(

lE

ElAem

For a uniform plane wave, the incident power density is given by

thus

51

Vector Effective Length• Vector effective length (or height) is a quantity used to

determine the voltage induced on the open-circuit

terminals of an antenna when a wave impinges on it. It is a far-field quantity.

ZA

b

a

VT

[m]),(ˆ),(ˆ),( llle r

ltagecircuit vo-open :

field electricincident :

ocT

i

VV

E

r

e

i

oc lEVrr

Example 2.6 : The electric field of a short dipole is given by

sin8

ˆr

lekIj

jkr

ina

Er

sin2

ˆ lle r

e

jkrina le

r

kIjEE

rr

4ˆˆE

52

Effective area & Directivity

)1(4or

4

2

2

RP

PAD

R

ADPAWP

t

rrt

rttrtr

204 R

PW t

If antenna #1 were isotropic, its radiated power density at a

distance R would be

where Pt is the total radiated power. Because of the directivity,

the actual power density becomes

The power collected by the antenna would be

204 R

DPDWW tt

tt

53

Effective area & Directivity(2)

rm

r

tm

t

A

D

A

D 00

)2(4 2R

P

PAD

t

rtr

r

r

t

t

A

D

A

D

If antenna #2 is used as a transmitter, 1 as a receiver, and the

medium is linear, passive and isotropic, one obtains

From (1) and (2),

Increasing the directivity of an antenna increases its effective

area:

r

rmtm

D

AA

0

If antenna #1 is isotropic,

54

Effective area & Directivity(3)

0

2

4DAem

43

2

8

3 22

0

r

rmtm

D

AA

4

2

00 rtmrrm DADA

For example, if antenna #2 is a short dipole, whose effective

area is 3λ2/8π and directivity is 1.5, one obtains

and

In general, maximum effective area of any antenna is related to its maximum directivity by

If there’re conduction-

dielectric loss, polarization

loss and mismatch:

22

0

2

|ˆˆ|)||1(4

awcdem eDA

55

Friis Transmission Equation• The Friis transmission equation relates the power

received to the power transmitted between two antennas separated by a distance R > 2D2/λλλλ.

22 4

),(

4

),(

R

DPe

R

GPW ttttttttt

t

4

),(2

rrrrr DeA

2

2

22

|ˆˆ|)4(

),(),(

4),( rt

trrrtttrttrrrrr

R

PDDeeWDeP

Power density at distance R from the transmitting antenna:

)ˆ,,,,,( ttcdtttt eDGP )ˆ,,,,,( rrcdrrrr eDGP

),( rr ),( tt

The effective area of the

receiving antenna:

The amount of power collected by the receiving antenna:

56

Friis Transmission Equation(2)

2

2

222 |ˆˆ|

)4(

),(),()||1)(||1( rt

rrrtttrtcdrcdt

t

r

R

DDee

P

P

2

2

2

|ˆˆ|)4(

),(),(rt

rrrtttrt

t

r

R

DDee

P

P

The ratio of the received to the input power:

or

rt

t

r GGRP

P00

2

4

For reflection and polarization-matched antennas aligned for

maximum directional radiation and reception:

57

Radar Cross Section

• Radar cross section or echo area (σσσσ) of a target is

defined as the area intercepting that amount of power

which, when scattered isotropically, produces at the

receiver a density which is equal to that scattered by the

actual target.

2

22

2

222

||

||4lim

||

||4lim4lim

i

s

Ri

s

Ri

s

RRR

W

WR

H

H

E

Er

r

r

r

[A/m] field magnetic scattered incident, :,

[V/m] field electric scattered incident, :,

][W/mdensity power scattered incident, :,

[m] targefrom distancen observatio :

][msection crossradar :

2

2

si

si

si WW

R

HH

EErr

rr

58

59

Radar Range Equation

Transmitting

antenna

)ˆ,,,,,( ttcdtttt eDGP

)ˆ,,,,,( rrcdrrrr eDGP

w

Receiving

antenna

Target R1

Incident wave

R2

Scattered wave

2

21

2

2 )4(

),(

4 RR

DPe

R

PW tttt

tc

s

• The radar range equation relates the power delivered to

the receiver load to the power transmitted by an antenna,

after it has been scattered by a target with a radar cross section of σσσσ.

2

1

2

1 4

),(

4

),(

R

DPe

R

GPWP ttttttttt

tc

The amount of captured power at

the target with the distance R1 from

the transmitting antenna:

The scattered power density:

60

Radar Range Equation(2)

2

2144

),(),(

RR

DDPeeWAP rrrtttt

rtsrr

2

2144

),(),(

RR

DDeeWA

P

P rrrtttrtsr

t

r

2

2

21

22 |ˆˆ|44

),(),()||1)(||1( rw

rrrtttrtcdrcdtsr

t

r

RR

DDeeWA

P

P

The amount of power delivered to the load:

2

21

00

44

RR

GG

P

P rt

t

r

Thus,

With polarization loss:

For reflection and polarization-matched antennas aligned for

maximum directional radiation and reception:

Brightness Temperature

• Energy radiated by an object can be

represented by an equivalent temperature

known as Brightness Temperature TB (K).

where

εεεε : emissivity (dimensionless) 0 ≤ εεεε ≤ 1

Tm : molecular (physical) temperature (K)

ΓΓΓΓ(θθθθ,,,,φφφφ) : reflection coefficient of the surface for the

polarization of wave

Example Ground 300 K

61

mmB TTT2

1),(),(

Antenna Temperature• Energy radiated by various sources appears

at antenna terminal as antenna temperature,

given by

where

TA : antenna temperature (effective noise

temperature of the antenna radiation resistance ; K)

G(θθθθ,,,,φφφφ) : gain (power) pattern of the antenna

62

2

0 0

2

0 0

sin),(

sin),(),(

ddG

ddGTT

B

A

Noise Power

• Assuming no losses or other contributions

between antenna & receiver, noise power

transferred to receiver:

where

Pr : antenna noise power (W)

k : Boltzmann’s constant (1.38×10-23 J/K)

TA : antenna temperature (K)

∆∆∆∆f : bandwidth (Hz)63

fkTP Ar

System Noise Power Model

• Assume antenna & transmission line are

maintained at certain temperature, and

transmission line is lossy, then the model

below can be used to include all

contributions.

64

Antenna Temperature (2)• Effective antenna temperature at receiver

terminals:

where

Ta : antenna temperature at receiver terminals (K)

TA : antenna noise temperature at antenna terminals (K)

TAP : antenna temperature at antenna terminals due to

physical temperature (K)

αααα : attenuation constant of transmission line (Np/m)

eA : thermal efficiency of antenna (dimensionless)

l : length of transmission line (m)

T0 : physical temperature of transmission line (K)

65

)1( 2

0

22 ll

AP

l

Aa eTeTeTT

PAAP TeT )1( 1

System Noise Power

• noise power transferred to receiver:

• If there’s thermal noise in receiver:

where

Ps : system noise power (W)

Ta : antenna noise temperature at receiver terminals

Tr : receiver noise temperature at receiver terminals

Ts =Ta + Tr : effective system noise temperature at

receiver terminals

66

fkTP ar

fkTfTTkP sras )(

Ex 2.16 Effective antenna temp = 150 K.

Antenna is maintained at 300 K and has

thermal efficiency 99%. It is connected to a

receiver through 10-m waveguide (loss = 0.13

dB/m, temp = 300 K) Find effective antenna

temperature at receiver terminals.

67

03.3300)199(.)1( 11 PAAP TeT

149.00.0149,dB/m)/8.68(Np/m)( l

K904.190

)1(30003.3150

)1(

)149(.2)149(.2)149(.2

2

0

22

eee

eTeTeTTll

AP

l

Aa